POST /api/v1/simplex/solve — Linear Programming Solver

The /api/v1/simplex/solve endpoint accepts a linear objective function and a set of linear constraints, then applies the Simplex algorithm to find the optimal solution. Along with the final tableau the response includes a per-iteration snapshot of every pivot step, and — for 2-variable problems — a graphical payload ready for plotting.

Request

POST /api/v1/simplex/solve

Content-Type: application/json

objective

list[float]

required

Coefficients of the objective function, one per decision variable. Must contain between 2 and 5 elements. The number of elements determines the number of decision variables (x1, x2, …, x5).Example: [5, 4, 3] represents 5x1 + 4x2 + 3x3.

goal

string

required

Optimisation direction. Accepted values: "max" or "min".

constraints

list[Constraint]

required

At least one linear constraint. Every constraint must have the same number of coefficients as objective.

Show Constraint object

coefficients

list[float]

required

Left-hand-side coefficients, one per decision variable (2–5 elements, must match the length of objective).

inequality

string

required

Relationship operator. One of "<=", ">=", or "=".

rhs

float

required

Right-hand-side constant of the constraint.

Response

A successful call returns a SimplexResponse object.

status

string

Outcome of the solve. One of "optimal", "unbounded", or "infeasible".

objective_value

float | null

The optimal value of the objective function. null when status is not "optimal".

variables

object | null

Map of variable names to their optimal values, e.g. {"x1": 3.0, "x2": 2.5, "x3": 0.0}. null when status is not "optimal".

iterations

integer | null

Total number of pivot iterations performed. null when status is not "optimal".

tableau_headers

list[string] | null

Column labels of the final simplex tableau (includes decision variables, slacks, and "RHS"). null when status is not "optimal".

tableau_rows

list[TableauRow] | null

Rows of the final simplex tableau. null when status is not "optimal".

Show TableauRow object

basic_variable

string

Name of the basic variable for this row (e.g. "x1", "s2").

values

list[float]

Numeric values across all tableau columns in the same order as tableau_headers.

message

string | null

Human-readable status message describing the result.

graphical

GraphicalData | null

Graphical representation of the feasible region. Only populated when the problem has exactly 2 decision variables; otherwise null.

Show GraphicalData object

status

string

Same as the top-level status field.

bounds

object

Axis bounds for the plot: x_min, x_max, y_min, y_max.

constraints

list[GraphLine]

Each constraint rendered as a line with a, b, c coefficients, sample points, a label, and the inequality type.

feasible_polygon

list[GraphPoint] | null

Vertices of the feasible region polygon in {x, y} format, ordered for drawing. null when infeasible.

vertices

list[GraphPoint]

All corner points of the feasible region evaluated during the graphical scan.

objective_line

GraphLine | null

The objective function isoprofit/isocost line at the optimal value, formatted the same way as constraint lines.

optimal_point

object | null

The optimal {x, y} point.

objective_value

float | null

Optimal objective value echoed in the graphical payload.

goal

string

"max" or "min" — echoes the request goal.

iteration_tableaux

list[IterationTableau] | null

Ordered list of tableau snapshots, one per pivot iteration.

Show IterationTableau object

iteration

integer

1-based iteration number.

entering

string | null

Name of the variable that entered the basis at this pivot step.

leaving

string | null

Name of the variable that left the basis at this pivot step.

tableau_headers

list[string]

Column labels at this iteration (identical to the final tableau_headers).

tableau_rows

list[TableauRow]

Tableau rows at this iteration (same structure as the final tableau_rows).

Example

Maximize 5x1 + 4x2 + 3x3 subject to:

6x1 + 4x2 + 2x3 ≤ 240
3x1 + 2x2 + 5x3 ≤ 270
5x1 + 6x2 + 5x3 ≤ 420

curl -X POST https://api.example.com/api/v1/simplex/solve \
  -H "Content-Type: application/json" \
  -d '{
    "objective": [5, 4, 3],
    "goal": "max",
    "constraints": [
      {"coefficients": [6, 4, 2], "inequality": "<=", "rhs": 240},
      {"coefficients": [3, 2, 5], "inequality": "<=", "rhs": 270},
      {"coefficients": [5, 6, 5], "inequality": "<=", "rhs": 420}
    ]
  }'

{
  "status": "optimal",
  "objective_value": 220.0,
  "variables": {
    "x1": 40.0,
    "x2": 0.0,
    "x3": 0.0
  },
  "iterations": 2,
  "tableau_headers": ["x1", "x2", "x3", "s1", "s2", "s3", "RHS"],
  "tableau_rows": [
    {"basic_variable": "x1", "values": [1.0, 0.667, 0.333, 0.167, 0.0, 0.0, 40.0]},
    {"basic_variable": "s2", "values": [0.0, 0.0, 4.0, -0.5, 1.0, 0.0, 150.0]},
    {"basic_variable": "s3", "values": [0.0, 2.667, 3.333, -0.833, 0.0, 1.0, 220.0]}
  ],
  "message": "Optimal solution found",
  "graphical": null,
  "iteration_tableaux": [
    {
      "iteration": 1,
      "entering": "x1",
      "leaving": "s1",
      "tableau_headers": ["x1", "x2", "x3", "s1", "s2", "s3", "RHS"],
      "tableau_rows": [
        {"basic_variable": "x1", "values": [1.0, 0.667, 0.333, 0.167, 0.0, 0.0, 40.0]},
        {"basic_variable": "s2", "values": [0.0, 0.0, 4.0, -0.5, 1.0, 0.0, 150.0]},
        {"basic_variable": "s3", "values": [0.0, 2.667, 3.333, -0.833, 0.0, 1.0, 220.0]}
      ]
    }
  ]
}

When the problem has exactly 2 decision variables, the graphical field in the response is populated with line data, feasible polygon vertices, and the optimal point — useful for rendering a 2-D plot on the client side.

All constraint coefficients arrays must have the same length as objective. Mismatched lengths return HTTP 422 Unprocessable Entity.

Health Check

GET /api/v1/simplex/health Returns a simple liveness confirmation for the Simplex service.

curl https://api.example.com/api/v1/simplex/health

{
  "status": "ok",
  "service": "simplex-optimizer-api"
}

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